a To write a quotient as a complex number, multiply the numerator and the denominator by the complex conjugate of the denominator.
B
b The complex conjugate of 1-i is 1+i.
A
a 8/17+15/17i
B
b 2+5i
Practice makes perfect
a Recall that the number pairs a+bi and a-bi are complex conjugates. To write the complex conjugate of a complex number, we only change the sign of the imaginary part. Let's do that for the denominator of our given expression now.
Denominator:& 5-3i
Complex Conjugate:& 5+3iThe product of complex conjugates is a real number. To write the given quotient as a complex number, we will multiply the numerator and the denominator by the complex conjugate of the denominator.
5+3i/5-3i*5+3i/5+3i
This process is also known as rationalizing the denominator. Doing so will simplify the quotient.
b Similarly, recall that the number pairs a+bi and a-bi are complex conjugates. Let's again write the complex conjugate of the denominator of the given expression now.
Denominator:& 1-i
Complex Conjugate:& 1+iThe product of complex conjugates is a real number. To write the given quotient as a complex number, we will multiply the numerator and the denominator by the complex conjugate of the denominator.
7+3i/1-i*1+i/1+i
This process is also known as rationalizing the denominator. Doing so will simplify the quotient.
